130_notes.dvi

27.5 Vibrational States

We have seen that the energy of a molecule has a minimum for some particular separation between
atoms. Thislooks just like a harmonic oscillator potentialfor small variations from the
minimum. The molecule can “vibrate” in this potential giving rise to a harmonic oscillator energy
spectrum.

We canestimate the energy of the vibrational levels. IfEe∼ ̄hω= ̄h

√

k

me, then crudely the

proton has the same spring constant

√

k≈Ee

√m

e

̄h.

Evib∼ ̄h

√

k

M

=

√

m

M

Ee∼

1

10

eV

Recalling that room temperature is about 401 eV, this is approximately thermal energy, infrared.
The energy levels are simply

E= (n+

1

2

) ̄hωvib

Complex molecules can have many different modes of vibration. Diatomic molecules have just one.

The graph below shows the energy spectrum of electrons knockedout of molecular hydrogen by UV
photons (photoelectric effect). Thedifferent peaks correspond to the vibrational state of
the final H+ 2 ion.

Can you calculate the number of vibrational modes for a molecule compose ofN >3 atoms.